Refined blowup criteria and nonsymmetric blowup of an aggregation equation

Abstract

We consider an aggregation equation in Rd, d ≥ 2, with fractional dissipation: ut + ∇ ṡ (u ∇ K * u) = - ν Λγ u, where ν ≥ 0, 0 < γ < 1, and K (x) = e- | x |. We prove a refined blowup criteria by which the global existence of solutions is controlled by its Lxq norm, for any frac(d, d - 1) ≤ q ≤ ∞. We prove the finite time blowup of solutions for a general class of nonsymmetric initial data. The argument presented works for both the inviscid case ν = 0 and the supercritical case ν > 0 and 0 < γ < 1. Additionally, we present new proofs of blowup which does not use free energy arguments. © 2008 Elsevier Inc. All rights reserved.